 reserve f,g for Function;
 reserve R for non empty reflexive RelStr;
 reserve R for non empty RelStr;
 reserve f for Function of the carrier of R, bool the carrier of R;

theorem Propk: :: k)
  for f being Function of the carrier of R, bool the carrier of R
  for x,y being Subset of R holds
    (Flip ff_0 f).x \/ (Flip ff_0 f).y c= (Flip ff_0 f).(x \/ y)
  proof
    let f be Function of the carrier of R, bool the carrier of R;
    let x,y be Subset of R;
AA: (Flip ff_0 f).(x \/ y) =
      { u where u is Element of R : f.u c= (x \/ y) } by FlipFF;
AB: (Flip ff_0 f).x =
      { u where u is Element of R : f.u c= x } by FlipFF;
AC: (Flip ff_0 f).y =
      { u where u is Element of R : f.u c= y } by FlipFF;
XX:   x c= x \/ y & y c= x \/ y by XBOOLE_1:7;
      let t be object;
      assume t in (Flip ff_0 f).x \/ (Flip ff_0 f).y; then
      per cases by XBOOLE_0:def 3;
      suppose t in (Flip ff_0 f).x; then
        consider u being Element of R such that
A1:     t = u & f.u c= x by AB;
        f.u c= (x \/ y) by A1,XX;
        hence thesis by A1,AA;
      end;
      suppose t in (Flip ff_0 f).y; then
        consider u being Element of R such that
A1:     t = u & f.u c= y by AC;
        f.u c= (x \/ y) by A1,XX;
        hence thesis by A1,AA;
    end;
  end;
